Project ForgeFluid Mechanics
Bernoulli's Equation & Flow
GATE 2007
Hard
Consider a steady incompressible flow through a channel, uniform with value u₀ at inlet section A. The velocity profile at downstream section B is: u=Vm(y/δ) for 0≤y≤δ, u=Vm for δ≤y≤H−δ, u=Vm(H−y)/δ for H−δ≤y≤H. The ratio Vm/u₀ is
- A1/[1−2(δ/H)]
- B1
- 1/[1−(δ/H)]
- D1/[1+(δ/H)]
Explanation
By continuity, Q_A=u₀H must equal Q_B=∫u(y)dy over 0 to H. Integrating the piecewise-linear profile: ∫u dy = 2×(Vmδ/2) + Vm(H−2δ) = Vm(H−δ). Setting u₀H=Vm(H−δ) gives Vm/u₀=1/[1−(δ/H)].
Source: GATE ME 2007 (2007)
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